Tea Cups

Place the 16 different combinations of cup/saucer in this 4 by 4 arrangement so that no row or column contains more than one cup or saucer of the same colour.

Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

Lesser Digits

How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

Colour Building

Age 11 to 14
Challenge Level

Why do this problem ?

This problem provides an opportunity for students to work systematically, and to discover the powerful technique of using smaller cases to predict and explain what will happen with larger cases.

Possible approach

The problem assumes some familiarity with Cuisenaire rods, so if students are not familiar with the different colours and lengths, it may be worth spending some time first exploring a set of rods or using the online environment.

"I wonder how many different ways we could combine white rods (1) and red rods (2) to make the same length as the orange rod (10) ..."

Give students some time to think about the challenge, and then share their thoughts. The following might emerge:
"There are going to be loads of different ways"
"How are we going to be able to make sure we don't miss any?"

If no-one has suggested it: "Perhaps we could work on a simpler version of the problem to see if that helps. Let's see how many ways we could make the pink rod (4) out of whites and reds."
Once students have agreed on the five ways that a pink can be made from whites and reds, invite them to find the number of ways to make light green (3), yellow (5), and dark green (6). Then encourage them to make a prediction for black (7) and then test it out.

It is important to set aside enough time for students to think about and appreciate why each answer is the sum of the previous two. To draw out this insight, you might suggest that students organise their work into two categories: solutions that start with a red (what's left?) and solutions that start with a white (what's left now?)

Key questions

Is there a way to work systematically to make sure you have found all the possibilities?

When we make the length of the pink rod, how many possibilities have a red on the left?
How many possibilities have a white on the left?

Can you use this idea of thinking separately about the solutions that begin with a red, and the ones that begin with a white, to help you explain the patterns that you find?

Possible extension

Students could try 1 Step 2 Step, which has the same mathematical structure but set in a different context.

Possible support

Focus on how to work systematically to find all the solutions for smaller cases before introducing the orange rod challenge.

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